This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A094728 #26 Aug 04 2025 11:22:17
%S A094728 1,4,3,9,8,5,16,15,12,7,25,24,21,16,9,36,35,32,27,20,11,49,48,45,40,
%T A094728 33,24,13,64,63,60,55,48,39,28,15,81,80,77,72,65,56,45,32,17,100,99,
%U A094728 96,91,84,75,64,51,36,19,121,120,117,112,105,96,85,72,57,40,21
%N A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.
%C A094728 (T(n,k) mod 4) <> 2, see A042965, A016825.
%C A094728 All numbers m occur A034178(m) times.
%C A094728 The row polynomials T(n,x) appear in the calculation of the column g.f.s of triangle A120070 (used to find the frequencies of the spectral lines of the hydrogen atom).
%H A094728 G. C. Greubel, <a href="/A094728/b094728.txt">Table of n, a(n) for n = 1..5050</a>
%F A094728 Row polynomials: T(n,x) = n^2*Sum_{m=0..n} x^m - Sum_{m=0..n} m^2*x^m = Sum_{k=0..n-1} T(n,k)*x^k, n >= 1.
%F A094728 T(n, k) = A004736(n,k)*A094727(n,k).
%F A094728 T(n, 0) = A000290(n).
%F A094728 T(n, 1) = A005563(n-1) for n>1.
%F A094728 T(n, 2) = A028347(n) for n>2.
%F A094728 T(n, 3) = A028560(n-3) for n>3.
%F A094728 T(n, 4) = A028566(n-4) for n>4.
%F A094728 T(n, n-1) = A005408(n).
%F A094728 T(n, n-2) = A008586(n-1) for n>1.
%F A094728 T(n, n-3) = A016945(n-2) for n>2.
%F A094728 T(n, n-4) = A008590(n-2) for n>3.
%F A094728 T(n, n-5) = A017329(n-3) for n>4.
%F A094728 T(n, n-6) = A008594(n-3) for n>5.
%F A094728 T(n, n-8) = A008598(n-2) for n>7.
%F A094728 T(A005408(k), k) = A000567(k).
%F A094728 Sum_{k=0..n} T(n, k) = A002412(n) (row sums).
%F A094728 From _G. C. Greubel_, Mar 12 2024: (Start)
%F A094728 Sum_{k=0..n-1} (-1)^k * T(n, k) = A000384(floor((n+1)/2)).
%F A094728 Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A128624(n).
%F A094728 Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = (1/2)*n*(n+1 - (-1)^n*cos(n*Pi/2)). (End)
%F A094728 G.f.: x*(1 - 3*x^2*y + x*(1 + y))/((1 - x)^3*(1 - x*y)^2). - _Stefano Spezia_, Aug 04 2025
%e A094728 n=3: T(3,x) = 9+8*x+5*x^2.
%e A094728 Triangle begins:
%e A094728 1;
%e A094728 4, 3;
%e A094728 9, 8, 5;
%e A094728 16, 15, 12, 7;
%e A094728 25, 24, 21, 16, 9;
%e A094728 36, 35, 32, 27, 20, 11;
%e A094728 49, 48, 45, 40, 33, 24, 13;
%e A094728 64, 63, 60, 55, 48, 39, 28, 15;
%e A094728 81, 80, 77, 72, 65, 56, 45, 32, 17;
%e A094728 ... etc. - _Philippe Deléham_, Mar 07 2013
%t A094728 Table[n^2 - k^2, {n,12}, {k,0,n-1}]//Flatten (* _Michael De Vlieger_, Nov 25 2015 *)
%o A094728 (Magma) [n^2-k^2: k in [0..n-1], n in [1..15]]; // _G. C. Greubel_, Mar 12 2024
%o A094728 (SageMath) flatten([[n^2-k^2 for k in range(n)] for n in range(1,16)]) # _G. C. Greubel_, Mar 12 2024
%Y A094728 Cf. A000290, A000567, A004736, A005408, A005563, A008586, A008590.
%Y A094728 Cf. A008594, A008598, A016825, A016945, A017329, A028347, A028560.
%Y A094728 Cf. A028566, A034178, A042965, A094727, A120070, A128624.
%Y A094728 Cf. A000384 (alternating row sums), A002412 (row sums).
%K A094728 nonn,tabl,easy
%O A094728 1,2
%A A094728 _Reinhard Zumkeller_, May 24 2004